A339949 a(n) is the greatest runlength in all n-sections of the infinite Fibonacci word A014675.
2, 3, 5, 6, 7, 3, 2, 12, 4, 4, 4, 4, 18, 2, 3, 6, 20, 5, 3, 2, 30, 4, 3, 4, 4, 9, 2, 3, 9, 4, 4, 3, 4, 47, 2, 3, 5, 10, 6, 3, 2, 15, 4, 4, 4, 4, 13, 2, 3, 7, 8, 5, 3, 2, 77, 4, 3, 5, 6, 8, 3, 2, 10, 4, 4, 3, 4, 24, 2, 3, 6, 78, 6, 3, 2, 22, 4, 3, 4, 4, 11, 2
Offset: 1
Examples
For n >= 1, r = 0..n, k >= 0, let A014675(n*k+r) denote the k-th term of the r-th n-section of A014675; i.e., (A014675(k)) = 212212122122121221212212212122122121221212212212122121... has runlengths 1,1,2,1,1,1,2,1,2,1,...; a(1) = 2. (A014675(2k)) = 22112211222122212221122112221222122211221122112221222... has runlengths 2,2,2,2,3,1,3,1,3,2,... (A014675(2k+1)) = 122212221122112211222122211221122112221222122211221... has runlengths 1,3,1,3,2,2,2,2,2,3,...; a(2) = 3. (A014675(3k)) = 22111222211122221122222112222211222211122221112222111... has runlengths 2,3,4,3,4,2,5,2,5,2,4,3,4,3,... (A014675(3k+1)) = 112222111222211122221112222111222211222221122221112... has runlengths 2,4,3,4,3,4,3,4,3,4,,5,2,4,3,... (A014675(3k+2)) = 222211222221122221112222111222211122221112222112222... has runlengths 4,2,5,2,4,3,4,3,4,3,4,3,4,2,...; a(3) = 5.
Links
- Gandhar Joshi, Table of n, a(n) for n = 1..10000 (terms 1..232 from Jeffrey Shallit).
- Dmitry Badziahin and Jeffrey Shallit, Badly approximable numbers, Kronecker's theorem, and diversity of Sturmian characteristic sequences, arXiv:2006.15842 [math.NT], 2020.
- Gandhar Joshi and Dan Rust, Monochromatic arithmetic progressions in the Fibonacci word, arXiv:2501.05830 [math.NT], 2025. See p. 9.
Programs
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Mathematica
r = (1 + Sqrt[5])/2; z = 4000; f[n_] := Floor[(n + 2) r] - Floor[(n+1) r]; (* A014675 *) t = Table[Max[Map[Length,Union[Split[Table [f[n m], {n, 0, Floor[z/m]}]]]]], {m, 1, 20}, {n, 1, m}]; Map[Max, t] (* A339949 *)
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PARI
phi = quadgen(5); g(n) = min(frac(n * phi), 1 - frac(n * phi)); a(n) = if (g(n) <= (1 / phi)^2, ceil((1 / phi) / g(n)), 2*ceil(((1 / phi) - g(n)) / g(2 * n))); \\ Gandhar Joshi, Jan 14 2025
Formula
From Gandhar Joshi, Jan 14 2025: (Start)
phi = the golden ratio. g(n) = min {n*phi mod 1, 1 - (n*phi mod 1)}.
If g(n) <= (phi)^(-2), a(n) = ceiling{((phi)^(-1))/g(n)};
otherwise, a(n) = 2*ceiling{((phi)^(-1)-g(n))/g(2n)}. (End)
Extensions
a(61) corrected by Jeffrey Shallit, Mar 23 2021
Comments